{"id":212,"date":"2025-04-09T17:28:32","date_gmt":"2025-04-09T17:28:32","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-1-systems-of-linear-equations-two-variables-college-algebra-2e-openstax\/"},"modified":"2025-09-03T14:16:59","modified_gmt":"2025-09-03T14:16:59","slug":"7-1-systems-of-linear-equations-two-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-1-systems-of-linear-equations-two-variables\/","title":{"raw":"7.1 Systems of Linear Equations: Two Variables","rendered":"7.1 Systems of Linear Equations: Two Variables"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_0bb1dacf-671b-4fe5-a5a4-e67ad9a48d54\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Solve systems of equations by graphing.<\/li>\r\n \t<li>Solve systems of equations by substitution.<\/li>\r\n \t<li>Solve systems of equations by addition.<\/li>\r\n \t<li>Identify inconsistent systems of equations containing two variables.<\/li>\r\n \t<li>Express the solution of a system of dependent equations containing two variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_1027\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1027\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-300x155.webp\" alt=\"\" width=\"300\" height=\"155\" \/> Figure 1. (credit: Thomas S\u00f8renes)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134188850\">A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.<\/p>\r\n\r\n<section id=\"fs-id1165133236395\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Introduction to Systems of Equations<\/h2>\r\n<p id=\"fs-id1165137448108\">In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A <strong><span id=\"term-00001\" data-type=\"term\">system of linear equations<\/span><\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.<\/p>\r\n<p id=\"fs-id1165137885053\">In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y &amp;=&amp; 15 \\\\ 3x-y &amp;=&amp; 5 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165135596356\">The <em data-effect=\"italics\">solution<\/em> to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2(4)+(7) &amp;=&amp; 15 &amp;&amp; \\text{True} \\\\ 3(4)-(7) &amp;=&amp; 5 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165132912573\">In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A <strong>consistent system<\/strong> of equations has at least one solution. A consistent system is considered to be an <strong>independent system<\/strong> if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a <strong>dependent system<\/strong> if the equations have the same slope and the same <em data-effect=\"italics\">y<\/em>-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.<\/p>\r\n<p id=\"fs-id1165135262630\">Another type of system of linear equations is an <strong>inconsistent system<\/strong>, which is one in which the equations represent two parallel lines. The lines have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Types of Linear Systems <\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThere are three types of systems of linear equations in two variables, and three types of solutions.\r\n<ul>\r\n \t<li>An <strong>independent system<\/strong> has exactly one solution pair\u00a0[latex] (x, y). [\/latex] The point where the two lines intersect is the only solution.<\/li>\r\n \t<li>An <strong>inconsistent system<\/strong> has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\r\n \t<li>A <strong>dependent system<\/strong> has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\r\n<\/ul>\r\nFigure 2 compares the graphical representations of each type of system.\r\n\r\n[caption id=\"attachment_1029\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1029\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-300x116.webp\" alt=\"\" width=\"300\" height=\"116\" \/> Figure 2[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.<\/strong>\r\n<ol>\r\n \t<li>Substitute the ordered pair into each equation in the system.<\/li>\r\n \t<li>Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Determining Whether an Ordered Pair Is a Solution to a System of Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine whether the ordered pair [latex] (5, 1) [\/latex] is a solution to the given system of equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+3y &amp;=&amp; 8 \\\\ 2x-9 &amp;=&amp; y \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Substitute the ordered pair [latex] (5, 1) [\/latex] into both equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} (5)+3(1) &amp;=&amp; 8 \\\\ 8 &amp;=&amp; 8 &amp;&amp; \\text{True} \\\\ 2(5)-9 &amp;=&amp; (1) \\\\ 1 &amp;=&amp; 1 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\nThe ordered pair [latex] (5, 1) [\/latex] satisfies both equations, so it is the solution to the system.\r\n<h3>Analysis<\/h3>\r\nWe can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines. See Figure 3.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1030\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1030\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-300x298.webp\" alt=\"\" width=\"300\" height=\"298\" \/> Figure 3[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine whether the ordered pair [latex] (8, 5) [\/latex] is a solution to the following system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 5x-4y &amp;=&amp; 20 \\\\ 2x+1 &amp;=&amp; 3y \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134573832\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Systems of Equations by Graphing<\/h2>\r\n<p id=\"fs-id1165137911750\">There are multiple methods of solving systems of linear equations. For a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">system of linear equations<\/span> in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Solving a System of Equations in Two Variables by Graphing<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations by graphing. Identify the type of system.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y &amp;=&amp; -8 \\\\ x-y &amp;=&amp; -1 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Solve the first equation for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y &amp;=&amp; -8 \\\\ y &amp;=&amp; -2x-8\\end{array} [\/latex]<\/p>\r\nSolve the second equation for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-y &amp;=&amp; -1 \\\\ y &amp;=&amp; x+1 \\end{array} [\/latex]<\/p>\r\nGraph both equations on the same set of axes as in Figure 4.\r\n\r\n[caption id=\"attachment_1031\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1031\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-300x240.webp\" alt=\"\" width=\"300\" height=\"240\" \/> Figure 4[\/caption]\r\n\r\nThe lines appear to intersect at the point [latex] (-3, -2). [\/latex] We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2(-3)+(-2) &amp;=&amp; -8 \\\\ -8 &amp;=&amp; -8 &amp;&amp; \\text{True} \\\\ (-3)-(-2) &amp;=&amp; -1 \\\\ -1 &amp;=&amp; -1 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\nThe solution to the system is the ordered pair [latex] (-3, -2), [\/latex] so the system is independent.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations by graphing.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x-5y &amp;=&amp; -25 \\\\ -4x+5y &amp;=&amp; 35 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">Q&amp;A<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165137443116\"><strong>Q: Can graphing be used if the system is inconsistent or dependent?<\/strong><\/p>\r\n<p id=\"fs-id1165133015238\"><em data-effect=\"italics\">A: Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135192374\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Systems of Equations by Substitution<\/h2>\r\n<p id=\"fs-id1165134361391\">Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the <strong>substitution method<\/strong>, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a system of two equations in two variables, solve using the substitution method.<\/strong>\r\n<ol>\r\n \t<li>Solve one of the two equations for one of the variables in terms of the other.<\/li>\r\n \t<li>Substitute the expression for this variable into the second equation, then solve for the remaining variable.<\/li>\r\n \t<li>Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.<\/li>\r\n \t<li>Check the solution in both equations.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Solving a System of Equations in Two Variables by Substitution<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations by substitution.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -x+y &amp;=&amp; -5 \\\\ 2x-5y &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, we will solve the first equation for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -x+y &amp;=&amp; -5 \\\\ y &amp;=&amp; x-5 \\end{array} [\/latex]<\/p>\r\nNow we can substitute the expression [latex] x-5 [\/latex] for y in the second equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x-5y &amp;=&amp; 1 \\\\ 2x-5(x-5) &amp;=&amp; 1 \\\\ 2x-5x+25 &amp;=&amp; 1 \\\\ -3x &amp;=&amp; -24 \\\\ x &amp;=&amp; 8 \\end{array} [\/latex]<\/p>\r\nNow, we substitute [latex] x=8 [\/latex] into the first equation and solve for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -(8)+y &amp;=&amp; -5 \\\\ y &amp;=&amp; 3 \\end{array} [\/latex]<\/p>\r\nOur solution is [latex] (8, 3). [\/latex]\r\n\r\nCheck the solution by substituting [latex] (8, 3) [\/latex] into both equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -x+y &amp;=&amp; -5 \\\\ -(8)+(3) &amp;=&amp; -5 &amp;&amp; \\text{True} \\\\ 2x-5y &amp;=&amp; 1 \\\\ 2(8)-5(3) &amp;=&amp; 1 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations by substitution.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x &amp;=&amp; y+3 \\\\ 4 &amp;=&amp; 3x-2y \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"eip-id3042099\"><strong>Q: Can the substitution method be used to solve any linear system in two variables?<\/strong><\/p>\r\n<p id=\"fs-id1165132973467\"><em data-effect=\"italics\">A: Yes, but the method works best if one of the equations contains a coefficient of 1 or \u20131 so that we do not have to deal with fractions.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165132972949\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Systems of Equations in Two Variables by the Addition Method<\/h2>\r\n<p id=\"fs-id1165131968519\">A third method of <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">solving systems of linear equations<\/span> is the <strong><span id=\"term-00010\" data-type=\"term\">addition method<\/span><\/strong>. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a system of equations, solve using the addition method.<\/strong>\r\n<ol>\r\n \t<li>Write both equations with <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-variables on the left side of the equal sign and constants on the right.<\/li>\r\n \t<li>Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.<\/li>\r\n \t<li>Solve the resulting equation for the remaining variable.<\/li>\r\n \t<li>Substitute that value into one of the original equations and solve for the second variable.<\/li>\r\n \t<li>Check the solution by substituting the values into the other equation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Solving a System by the Addition Method<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the given system of equations by addition.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+2y &amp;=&amp; -1 \\\\ -x+y &amp;=&amp; 3 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Both equations are already set equal to a constant. Notice that the coefficient of <em>x<\/em> in the second equation, \u20131, is the opposite of the coefficient of <em>x<\/em> in the first equation, 1. We can add the two equations to eliminate <em>x<\/em> without needing to multiply by a constant.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+2y &amp;=&amp; -1 \\\\ -x+y &amp;=&amp; 3 \\\\ \\hline 3y &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\nNow that we have eliminated x, we can solve the resulting equation for <em>y<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3y &amp;=&amp; 2 \\\\ y &amp;=&amp; \\frac{2}{3} \\end{array} [\/latex]<\/p>\r\nThen, we substitute this value for <em>y<\/em> into one of the original equations and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -x+y &amp;=&amp; 3 \\\\ -x+\\frac{2}{3} &amp;=&amp; 3 \\\\ -x &amp;=&amp; 3-\\frac{2}{3} \\\\ -x &amp;=&amp; \\frac{7}{3} \\\\ x &amp;=&amp; -\\frac{7}{3} \\end{array} [\/latex]<\/p>\r\nThe solution to this system is [latex] \\left(-\\frac{7}{3}, \\frac{2}{3}\\right). [\/latex]\r\n\r\nCheck the solution in the first equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+2y &amp;=&amp; -1 \\\\ \\left(-\\frac{7}{3}\\right)+2\\left(\\frac{2}{3}\\right) &amp;=&amp; -1 \\\\ -\\frac{7}{3}+\\frac{4}{3} &amp;=&amp; -1 \\\\ -\\frac{3}{3} &amp;=&amp; -1 \\\\ -1 &amp;=&amp; -1 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nWe gain an important perspective on systems of equations by looking at the graphical representation. See Figure 5 to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution.\r\n\r\n[caption id=\"attachment_1032\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1032\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-300x236.webp\" alt=\"\" width=\"300\" height=\"236\" \/> Figure 5[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165137760592\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135190076\" data-type=\"commentary\">\r\n<div id=\"Figure_09_01_004\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Using the Addition Method When Multiplication of One Equation Is Required<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the given system of equations by the addition method.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3x+5y &amp;=&amp; -11 \\\\ x-2y &amp;=&amp; 11 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Adding these equations as presented will not eliminate a variable. However, we see that the first equation has 3x in it and the second equation has <em>x<\/em>. So if we multiply the second equation by -3, the <em data-effect=\"italics\">x<\/em>-terms will add to zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2y &amp;=&amp; 11 \\\\ -3(x-2y) &amp;=&amp; -3(11) &amp;&amp; \\text{Multiply both sides by } -3. \\\\ -3x+6y &amp;=&amp; -33 &amp;&amp; \\text{Use the distributive property.} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165134567942\">Now, let\u2019s add them.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3x+5y &amp;=&amp; -11 \\\\ -3x+6y &amp;=&amp; -33 \\\\ \\hline 11y &amp;=&amp; -44 \\\\ y &amp;=&amp; -4 \\end{array} [\/latex]<\/p>\r\nFor the last step, we substitute [latex] y=-4 [\/latex] into one of the original equations and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 3x+5y &amp;=&amp; -11 \\\\ 3x+5(-4) &amp;=&amp; -11 \\\\ 3x-20 &amp;=&amp; -11 \\\\ 3x &amp;=&amp; 9 \\\\ x &amp;=&amp; 3 \\end{array} [\/latex]<\/p>\r\nOur solution is the ordered pair [latex] (3, -4). [\/latex] See Figure 6. Check the solution in the original second equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x-2y &amp;=&amp; 11 \\\\ (3)-2(-4) &amp;=&amp; 3+8 \\\\ 11 &amp;=&amp; 11 &amp;&amp; \\text{True} \\end{array} [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_1033\" align=\"aligncenter\" width=\"285\"]<img class=\"size-medium wp-image-1033\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-285x300.webp\" alt=\"\" width=\"285\" height=\"300\" \/> Figure 6[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of equations by addition.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x-7y &amp;=&amp; 2 \\\\ 3x+y &amp;=&amp; -20 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Using the Addition Method When Multiplication of Both Equations Is Required<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the given system of equations in two variables by addition.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll}2x+3y &amp;=&amp; -16 \\\\ 5x-10y &amp;=&amp; 30 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>One equation has 2x and the other has 5x. The least common multiple is 10x so we will have to multiply both equations by a constant in order to eliminate one variable. Let\u2019s eliminate <em>x<\/em> by multiplying the first equation by [latex] -5 [\/latex] and the second equation by [latex] 2. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -5(2x+3y) &amp;=&amp; -5(-16) \\\\ -10x-15y &amp;=&amp; 80 \\\\ 2(5x-10y) &amp;=&amp; 2(30) \\\\ 10x-20y &amp;=&amp; 60 \\end{array} [\/latex]<\/p>\r\nThen, we add the two equations together.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -10x-15y &amp;=&amp; 80 \\\\ 10x-20y &amp;=&amp; 60 \\\\ \\hline -35y &amp;=&amp; 140 \\\\ y &amp;=&amp; -4 \\end{array} [\/latex]<\/p>\r\nSubstitute [latex] y=-4 [\/latex] into the original first equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+3(-4) &amp;=&amp; -16 \\\\ 2x-12 &amp;=&amp; -16 \\\\ 2x &amp;=&amp; -4 \\\\ x &amp;=&amp; -2 \\end{array} [\/latex]<\/p>\r\nThe solution is [latex]- (-2, -4). [\/latex] Check it in the other equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 5x-10y &amp;=&amp; 30 \\\\ 5(-2)-10(-4) &amp;=&amp; 30 \\\\ -10+40 &amp;=&amp; 30 \\\\ 30 &amp;=&amp; 30 \\end{array} [\/latex]<\/p>\r\nSee Figure 7.\r\n\r\n[caption id=\"attachment_1034\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1034\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-300x230.webp\" alt=\"\" width=\"300\" height=\"230\" \/> Figure 7[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Using the Addition Method in Systems of Equations Containing Fractions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the given system of equations in two variables by addition.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} \\frac{x}{3}+\\frac{y}{6} &amp;=&amp; 3 \\\\ \\frac{x}{2}-\\frac{y}{4} &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First clear each equation of fractions by multiplying both sides of the equation by the least common denominator.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 6\\left(\\frac{x}{3}+\\frac{y}{6}\\right) &amp;=&amp; 6(3) \\\\ 2x+y &amp;=&amp; 18 \\\\ 4\\left(\\frac{x}{2}-\\frac{y}{4}\\right) &amp;=&amp; 4(1) \\\\ 2x-y &amp;=&amp; 4 \\end{array} [\/latex]<\/p>\r\nNow multiply the second equation by [latex] -1 [\/latex] so that we can eliminate the <em data-effect=\"italics\">x<\/em>-variable.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -1(2x-y) &amp;=&amp; -1(4) \\\\ -2x+y &amp;=&amp; -4 \\end{array} [\/latex]<\/p>\r\nAdd the two equations to eliminate the <em data-effect=\"italics\">x<\/em>-variable and solve the resulting equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+y &amp;=&amp; 18 \\\\ -2x+y &amp;=&amp; -4 \\\\ \\hline 2y &amp;=&amp; 14 \\\\ y &amp;=&amp; 7 \\end{array} [\/latex]<\/p>\r\nSubstitute [latex] y=7 [\/latex] into the first equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+(7) &amp;=&amp; 18 \\\\ 2x &amp;=&amp; 11 \\\\ x &amp;=&amp; \\frac{11}{2} \\\\ &amp;=&amp; 5.5 \\end{array} [\/latex]<\/p>\r\nThe solution is [latex] \\left(\\frac{11}{2}, 7\\right). [\/latex] Check it in the other equation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} \\frac{x}{2}-\\frac{y}{4} &amp;=&amp; 1 \\\\ \\frac{\\frac{11}{2}}{2}-\\frac{7}{4} &amp;=&amp; 1 \\\\ \\frac{11}{4}-\\frac{7}{4} &amp;=&amp; 1 \\\\ \\frac{4}{4} &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the system of equations by addition.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2x+3y &amp;=&amp; 8 \\\\ 3x+5y &amp;=&amp; 10 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135462954\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Two Variables<\/h2>\r\n<p id=\"fs-id1165135462960\">Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">inconsistent system<\/span> consists of parallel lines that have the same slope but different <em>y<\/em>-intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex] 12=0. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Solving an Inconsistent System of Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x &amp;=&amp; 9-2y \\\\ x+2y &amp;=&amp; 13 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can approach this problem in two ways. Because one equation is already solved for <em>x<\/em>, the most obvious step is to use substitution.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+2y &amp;=&amp; 13 \\\\ (9-2y)+2y &amp;=&amp; 13 \\\\ 9+0y &amp;=&amp; 13 \\\\ 9 &amp;=&amp; 13 \\end{array} [\/latex]<\/p>\r\nClearly, this statement is a contradiction because [latex] 9\\not=13. [\/latex] Therefore, the system has no solution.\r\n\r\nThe second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x &amp;=&amp; 9-2y \\\\ 2y &amp;=&amp; -x+9 \\\\ y &amp;=&amp; -\\frac{1}{2}x+\\frac{9}{2} \\end{array} [\/latex]<\/p>\r\nWe then convert the second equation expressed to slope-intercept form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+2y &amp;=&amp; 13 \\\\ 2y &amp;=&amp; -x+13 \\\\ y &amp;=&amp; -\\frac{1}{2}x+\\frac{13}{2} \\end{array} [\/latex]<\/p>\r\nComparing the equations, we see that they have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. Therefore, the lines are parallel and do not intersect.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; -\\frac{1}{2}x+\\frac{9}{2} \\\\ y &amp;=&amp; -\\frac{1}{2}+\\frac{13}{2} \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nWriting the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure 8.\r\n\r\n[caption id=\"attachment_1037\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1037\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-300x213.webp\" alt=\"\" width=\"300\" height=\"213\" \/> Figure 8[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165137701571\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165134116945\" data-type=\"commentary\">\r\n<div id=\"Figure_09_01_007\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations in two variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 2y-2x &amp;=&amp; 2 \\\\ 2y-2x &amp;=&amp; 6 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135189716\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Expressing the Solution of a System of Dependent Equations Containing Two Variables<\/h2>\r\n<p id=\"fs-id1165133200582\">Recall that a <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">dependent system<\/span> of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as [latex] 0=0. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Finding a Solution to a Dependent System of Linear Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a solution to the system of equations using the addition method.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+3y &amp;=&amp; 2 \\\\ 3x+9y &amp;=&amp; 6 \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let\u2019s focus on eliminating <em>x<\/em>. If we multiply both sides of the first equation by [latex] -3, [\/latex] then we will be able to eliminate the <em>x<\/em>-variable.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+3y &amp;=&amp; 2 \\\\ (-3)(x+3y) &amp;=&amp; (-3)(2) \\\\ -3x-9y &amp;=&amp; -6 \\end{array} [\/latex]<\/p>\r\nNow add the equations.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} -3x-9y &amp;=&amp; -6 \\\\ + \\ \\ \\ 3x+9y &amp;=&amp; 6 \\\\ \\hline 0 &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\nWe can see that there will be an infinite number of solutions that satisfy both equations.\r\n<h3>Analysis<\/h3>\r\nIf we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let\u2019s look at what happens when we convert the system to slope-intercept form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} x+3y &amp;=&amp; 2 \\\\ 3y &amp;=&amp; -x+2 \\\\ y &amp;=&amp; -\\frac{1}{3}x+\\frac{2}{3} \\\\ 3x+9y &amp;=&amp; 6 \\\\ 9y &amp;=&amp; -3x+6 \\\\ y &amp;=&amp; -\\frac{3}{9}x+\\frac{6}{9} \\\\ y &amp;+&amp; -\\frac{1}{3}x+\\frac{2}{3} \\end{array} [\/latex]<\/p>\r\nSee Figure 9. Notice the results are the same. The general solution to the system is [latex] \\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right). [\/latex]\r\n\r\n[caption id=\"attachment_1039\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1039\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-300x254.webp\" alt=\"\" width=\"300\" height=\"254\" \/> Figure 9[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165133299011\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165133188520\" data-type=\"commentary\">\r\n<div id=\"Figure_09_01_008\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve the following system of equations in two variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y-2x &amp;=&amp; 5 \\\\ -3y+6x &amp;=&amp; -15 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135559421\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Systems of Equations to Investigate Profits<\/h2>\r\n<p id=\"fs-id1165135559426\">Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer\u2019s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation [latex] R=xp, [\/latex] where [latex] x= [\/latex] quantity and [latex] p= [\/latex] price. The revenue function is shown in orange in Figure 10.<\/p>\r\n<p id=\"fs-id1165135459707\">The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in Figure 10. The x-axis represents quantity in hundreds of units. The <em data-effect=\"italics\">y<\/em>-axis represents either cost or revenue in hundreds of dollars.<\/p>\r\n&nbsp;\r\n<div id=\"Figure_09_01_009\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n\r\n[caption id=\"attachment_1040\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1040\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-300x212.webp\" alt=\"\" width=\"300\" height=\"212\" \/> Figure 10[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135449745\">The point at which the two lines intersect is called the <strong>break-even point<\/strong>. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.<\/p>\r\n<p id=\"fs-id1165135449753\">The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The <strong>profit function<\/strong> is the revenue function minus the cost function, written as [latex] P(x)=R(x)-C(x). [\/latex] Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Finding the Break-Even Point and the Profit Function Using Substitution<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the cost function [latex] C(x)=0.85x+35,000 [\/latex] and the revenue function [latex] R(x)=1.55x, [\/latex] find the break-even point and the profit function.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Write the system of equations using y to replace function notation.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} y &amp;=&amp; 0.85x+35,000 \\\\ y &amp;=&amp; 1.55x \\end{array} [\/latex]<\/p>\r\nSubstitute the expression [latex] 0.85x+35,000 [\/latex] from the first equation into the second equation and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 0.85x+35,000 &amp;=&amp; 1.55x \\\\ 35,000 &amp;=&amp; 0.7x \\\\ 50,000 &amp;=&amp; x \\end{array} [\/latex]<\/p>\r\nThen, we substitute [latex] x=50,000 [\/latex] into either the cost function or the revenue function.\r\n<p style=\"text-align: center;\">[latex] 1.55(50,000)=77,500 [\/latex]<\/p>\r\nThe break-even point is [latex] (50,000, 77,500). [\/latex]\r\n\r\nThe profit function is found using the formula [latex] P(x)=R(x)-C(x). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} P(x) &amp;=&amp; 1.55x-(0.85x+35,000) \\\\ &amp;=&amp; 0.7x-35,000 \\end{array} [\/latex]<\/p>\r\nThe profit function is [latex] P(x)=0.7x-35,000. [\/latex]\r\n<h3>Analysis<\/h3>\r\nThe cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11.\r\n\r\n[caption id=\"attachment_1041\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1041\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-300x211.webp\" alt=\"\" width=\"300\" height=\"211\" \/> Figure 11[\/caption]\r\n\r\nWe see from the graph in Figure 12 that the profit function has a negative value until [latex] x=50,000, [\/latex] when the graph crosses the <em data-effect=\"italics\">x<\/em>-axis. Then, the graph emerges into positive <em data-effect=\"italics\">y<\/em>-values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.\r\n\r\n[caption id=\"attachment_1042\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1042\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-300x168.webp\" alt=\"\" width=\"300\" height=\"168\" \/> Figure 12[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165134040438\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165131987013\" data-type=\"commentary\">\r\n<div id=\"Figure_09_01_010\" class=\"os-figure\"><\/div>\r\n<div id=\"Figure_09_01_011\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Writing and Solving a System of Equations in Two Variables<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe cost of a ticket to Elitch Gardens is for [latex] \\$25.00 [\/latex] children and [latex] \\$50.00 [\/latex] for adults. On a certain day, attendance at Elitch Gardens is [latex] 2,000 [\/latex] and the total gate revenue is [latex] \\$70,000. [\/latex]How many children and how many adults bought tickets?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Let <em data-effect=\"italics\">c<\/em> = the number of children and <em data-effect=\"italics\">a<\/em> = the number of adults in attendance.\r\n\r\nThe total number of people is 2,000. We can use this to write an equation for the number of people at the circus that day.\r\n<p style=\"text-align: center;\">[latex] c+a=2,000 [\/latex]<\/p>\r\nThe revenue from all children can be found by multiplying [latex] \\$25.00 [\/latex] by the number of children, [latex] 25c. [\/latex] The revenue from all adults can be found by multiplying [latex] \\$50.000 [\/latex] by the number of adults, [latex] 50a. [\/latex] The total revenue is [latex] \\$70,000. [\/latex] We can use this to write an equation for the revenue.\r\n<p style=\"text-align: center;\">[latex] 25c+50a=70,000 [\/latex]<\/p>\r\nWe now have a system of linear equations in two variables.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} c+a &amp;=&amp; 2,000 \\\\ 25c+50a &amp;=&amp; 70,000 \\end{array} [\/latex]<\/p>\r\nIn the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either <em>c<\/em> or <em>a<\/em>. We will solve for <em>a<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} c+a &amp;=&amp; 2,000 \\\\ a &amp;=&amp; 2,000-c \\end{array} [\/latex]<\/p>\r\nSubstitute the expression [latex] 2,000-c [\/latex] in the second equation for <em>a<\/em> and solve for <em>c<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 25x+50(2,000-c) &amp;=&amp; 70,000 \\\\ 25c+100,000-50c &amp;=&amp; 70,000 \\\\ -25c &amp;=&amp; -30,000 \\\\ c &amp;=&amp; 1,200 \\end{array} [\/latex]<\/p>\r\nSubstitute [latex] c=1,200 [\/latex] into the first equation to solve for <em>a<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} 1,200+a &amp;=&amp; 2,000 \\\\ a &amp;=&amp; 800 \\end{array} [\/latex]<\/p>\r\nWe find that 1,200 children and 800 adults bought tickets to Elitch Gardens that day.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMeal tickets at Elitch Gardens cost [latex] \\$4.00 [\/latex] for children and [latex] \\$12.00 [\/latex] for adults. If 1,650 meal tickets were bought for a total of [latex] \\$14,200 [\/latex] how many children and how many adults bought meal tickets?\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with systems of linear equations.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=HxhacvH49o8\">Solving Systems of Equations Using Substitution<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=ova8GSmPV4o\">Solving Systems of Equations Using Elimination<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=GTolISAXveg\">Applications of Systems of Equations<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.1 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135186955\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135186958\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165133393400\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133393402\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133393400-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can a system of linear equations have exactly two solutions? Explain why or why not.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133393410\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133393411\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company\u2019s profit margins.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133393418\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133393419\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133393418-solution\">3<\/a><span class=\"os-divider\">. <\/span>If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133305366\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133305367\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133305372\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133305374\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133305372-solution\">5<\/a><span class=\"os-divider\">. <\/span>Given a system of equations, explain at least two different methods of solving that system.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165133232429\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165133232434\">For the following exercises, determine whether the given ordered pair is a solution to the system of equations.<\/p>\r\n\r\n<div id=\"fs-id1165133232439\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133232440\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll}5x-y &amp;=&amp; 4 \\\\ x+6y &amp;=&amp; 2 \\end{array} [\/latex] and [latex] (4, 0) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165132934456\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132934457\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132934456-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -3x-5y &amp;=&amp; 13 \\\\ -x+4y &amp;=&amp; 10 \\end{array} [\/latex] and [latex] (-6, 1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133294269\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133294270\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x+7y &amp;=&amp; 1 \\\\ 2x+4y &amp;=&amp; 0 \\end{array} [\/latex] and [latex] (2, 3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135255522\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135255523\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135255522-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -2x+5y &amp;=&amp; 7 \\\\ 2x+9y &amp;=&amp; 7 \\end{array} [\/latex] and [latex] (-1, 1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134350008\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134350009\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+8y &amp;=&amp; 43 \\\\ 3x-2y &amp;=&amp; -1 \\end{array} [\/latex] and [latex] (3, 5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134556458\">For the following exercises, solve each system by substitution.<\/p>\r\n\r\n<div id=\"fs-id1165134556461\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134556462\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134556461-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+3y &amp;=&amp; 5 \\\\ 2x+3y &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137666024\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137666025\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-2y &amp;=&amp; 18 \\\\ 5x+10y &amp;=&amp; -10 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134254239\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134254240\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134254239-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 4x+2y &amp;=&amp; -10\\\\ 3x+9y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133265018\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133265020\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2x+4y &amp;=&amp; -3.8 \\\\ 9x-5y &amp;=&amp; 1.3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134171257\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134171258\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134171257-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -2x+3y &amp;=&amp; 1.2 \\\\ -3x-6y &amp;=&amp; 1.8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137696015\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137696016\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x-0.2y &amp;=&amp; 1 \\\\ -10x+2y &amp;=&amp; 5 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133447806\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133447807\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133447806-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x+5y &amp;=&amp; 9 \\\\ 30x+50y &amp;=&amp; -90 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133370753\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133370754\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -3x+y &amp;=&amp; 2 \\\\ 12x-4y &amp;=&amp; -8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134162139\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134162140\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134162139-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{2}x+\\frac{1}{3}y &amp;=&amp; 16 \\\\ \\frac{1}{6}x+\\frac{1}{4}y &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134319658\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134319659\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -\\frac{1}{4}x+\\frac{3}{2}y &amp;=&amp; 11 \\\\ -\\frac{1}{8}x+\\frac{1}{3}y &amp;=&amp; 3\\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134568907\">For the following exercises, solve each system by addition.<\/p>\r\n\r\n<div id=\"fs-id1165134568910\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134568911\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134568910-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -2x+5y &amp;=&amp; -42 \\\\ 7x+2y &amp;=&amp; 30 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135407254\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135407255\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 6x-5y &amp;=&amp; -34 \\\\ 2x+6y &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134554922\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134554923\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134554922-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-y &amp;=&amp; -2.6 \\\\ -4x-6y &amp;=&amp; 1.4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134357764\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134357765\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 7x-2y &amp;=&amp; 3 \\\\ 4x+5y &amp;=&amp; 3.25 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132919470\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132919471\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132919470-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x+2y &amp;=&amp; -1 \\\\ 5x-10y &amp;=&amp; 6 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133109448\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133109449\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 7x+6y &amp;=&amp; 2 \\\\ -28x-24y &amp;=&amp; -8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132924229\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132924230\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132924229-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{5}{6}x+\\frac{1}{4}y &amp;=&amp; 0 \\\\ \\frac{1}{8}x-\\frac{1}{2}y &amp;=&amp; -\\frac{43}{120} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132931032\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132931033\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{3}x+\\frac{1}{9}y &amp;=&amp; \\frac{2}{9} \\\\ -\\frac{1}{2}x+\\frac{4}{5}y &amp;=&amp; -\\frac{1}{3} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135183918\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135183919\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135183918-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -0.2x+0.4y &amp;=&amp; 0.6 \\\\ x-2y &amp;=&amp; -3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131991291\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131991292\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -0.1x+0.2y &amp;=&amp; 0.6 \\\\ 5x-10y &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134278608\">For the following exercises, solve each system by any method.<\/p>\r\n\r\n<div id=\"fs-id1165134278611\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134278612\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134278611-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x+9y &amp;=&amp; 16 \\\\ x+2y &amp;=&amp; 4 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133172708\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133172709\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 6x-8y &amp;=&amp; -0.6 \\\\ 3x+2y &amp;=&amp; 0.9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134430261\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134430262\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134430261-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 5x-2y &amp;=&amp; 2.25 \\\\ 7x-4y &amp;=&amp; 3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135705758\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135705759\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x-\\frac{5}{12}y &amp;=&amp; -\\frac{55}{12} \\\\ -6x+\\frac{5}{2}y &amp;=&amp; \\frac{55}{2} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134329672\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134329673\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134329672-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 7x-4y &amp;=&amp; \\frac{7}{6} \\\\ 2x+4y &amp;=&amp; \\frac{1}{3} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134272660\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134272661\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x+6y &amp;=&amp; 11 \\\\ 2x+4y &amp;=&amp; 9 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132939103\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132939104\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132939103-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{7}{3}x-\\frac{1}{6}y &amp;=&amp; 2 \\\\ -\\frac{21}{6}x+\\frac{3}{12}y &amp;=&amp; -3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131893549\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131893550\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} \\frac{1}{2}x+\\frac{1}{3}y &amp;=&amp; \\frac{1}{3} \\\\ \\frac{3}{2}x+\\frac{1}{4}y &amp;=&amp; -\\frac{1}{8} \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135299787\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135299788\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135299787-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 2.2x+1.3y &amp;=&amp; -0.1 \\\\ 4.2x+4.2y &amp;=&amp; 2.1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135702539\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135702540\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.1x+0.2y &amp;=&amp; 2 \\\\ 0.35x-0.3y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135518201\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135518207\">For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.<\/p>\r\n\r\n<div id=\"fs-id1165135518212\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135518213\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135518212-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-y &amp;=&amp; 0.6 \\\\ x-2y &amp;=&amp; 1.3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165134279206\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134279207\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -x+2y &amp;=&amp; 4 \\\\ 2x-4y &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132957245\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132957246\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132957245-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+2y &amp;=&amp; 7 \\\\ 2x+6y &amp;=&amp; 12 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133074995\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133074996\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-5y &amp;=&amp; 7 \\\\ x-2y &amp;=&amp; 3 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133243607\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133243608\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133243607-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 3x-2y &amp;=&amp; 5 \\\\ -9x+6y &amp;=&amp; -15 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165133290332\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135388295\">For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.<\/p>\r\n\r\n<div id=\"fs-id1165135388300\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135388301\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.1x+0.2y &amp;=&amp; 0.3 \\\\ -0.3x+0.5y &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137836999\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137837000\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137836999-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -0.01x+0.12y &amp;=&amp; 0.62 \\\\ 0.15x+0.20y &amp;=&amp; 0.52 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134035993\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134035994\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.5x+0.3y &amp;=&amp; 4 \\\\ 0.25x-0.9y &amp;=&amp; 0.46 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134361357\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134361358\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134361357-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} 0.15x+0.27y &amp;=&amp; 0.39 \\\\ -0.34x+0.56y &amp;=&amp; 1.8 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134485557\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134485558\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} -0.71x+0.92y &amp;=&amp; 0.13 \\\\ 0.83x+0.05y &amp;=&amp; 2.1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134113096\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165134113102\">For the following exercises, solve each system in terms of [latex] A, B, C, D, E, [\/latex] and [latex] F [\/latex] where [latex] A-F [\/latex] are nonzero numbers. Note that [latex] A\\not=B [\/latex] and [latex] AE\\not=BD. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165132923031\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132923032\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132923031-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+y &amp;=&amp; A \\\\ x-y &amp;=&amp; B \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135621951\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134385523\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} x+Ay &amp;=&amp; 1 \\\\ x+By &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132972781\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132972782\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132972781-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} Ax+y &amp;=&amp; 0 \\\\ Bx+y &amp;=&amp; 0 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137848824\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137848825\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} Ax+By &amp;=&amp; C \\\\ x+y &amp;=&amp; 1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133210727\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133210728\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133210727-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] \\begin{array}{rll} Ax+By &amp;=&amp; C \\\\ Dx+Ey &amp;=&amp; F \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135702580\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1165135702585\">For the following exercises, solve for the desired quantity.<\/p>\r\n\r\n<div id=\"fs-id1165135702588\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135702589\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>A stuffed animal business has a total cost of production [latex] C=12x+30 [\/latex] and a revenue function [latex] R=20x. [\/latex] Find the break-even point.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135486976\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135486977\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135486976-solution\">57<\/a><span class=\"os-divider\">. <\/span>An Ethiopian restaurant has a cost of production [latex] C(x)=11x+120 [\/latex] and a revenue function [latex] R(x)=5x. [\/latex] When does the company start to turn a profit?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133315062\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133315063\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>A cell phone factory has a cost of production [latex] C(x)=150x+10,000 [\/latex] and a revenue function [latex] R(x)=200x. [\/latex] What is the break-even point?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133249125\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133249126\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133249125-solution\">59<\/a><span class=\"os-divider\">. <\/span>A musician charges [latex] C(x)=64x+20,000 [\/latex] where <em>x<\/em> is the total number of attendees at the concert. Red Rocks Ampitheatre charges $80 per ticket. After how many people buy tickets does Red Rocks Ampitheatre break even, and what is the value of the total tickets sold at that point?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134231536\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134231537\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>A guitar factory has a cost of production [latex] C(x)=75x+50,000. [\/latex] If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165131863006\">For the following exercises, use a system of linear equations with two variables and two equations to solve.<\/p>\r\n\r\n<div id=\"fs-id1165131863011\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131863012\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131863011-solution\">61<\/a><span class=\"os-divider\">. <\/span>Find two numbers whose sum is 28 and difference is 13.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131863020\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131863021\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131863026\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131863027\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131863026-solution\">63<\/a><span class=\"os-divider\">. <\/span>The startup cost for a restaurant in Aurora, CO, is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199413\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199414\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199420\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199422\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134199420-solution\">65<\/a><span class=\"os-divider\">. <\/span>A total of 1,595 first- and second-year CCA students gathered at a pep rally. The number of first-years exceeded the number of second-years by 15. How many students from each year group were in attendance?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199432\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199433\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>276 CCA students enrolled in a College Algebra class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199439\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199440\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134199439-solution\">67<\/a><span class=\"os-divider\">. <\/span>There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134254378\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134254379\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>A jeep and a pickup truck enter I 70 running east-west at the same exit heading in opposite directions. The jeep entered I 70 30 minutes before the pickup did, and traveled 7 mph slower than the pickup. After 2 hours from the time the pickup entered I 70, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control and retained the same speed.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134254387\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134254388\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134254387-solution\">69<\/a><span class=\"os-divider\">. <\/span>If a CCA Chemistry Professor mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many\r\ngallons of 10% and 60% solutions were mixed?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134254398\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134254399\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>An investor earned triple the profits of what they earned last year. If they made $500,000.48 total for both years, how much did the investor earn in profits each year?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133227637\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133227638\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133227637-solution\">71<\/a><span class=\"os-divider\">. <\/span>An investor invested 1.1 million dollars into two land investments. On the first investment, the Aurora Highlands, her return was a 110% increase on the money she invested. On the second investment, Westerley Creek, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133227650\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133227651\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the other that pays [latex] 2 \\frac{7}{8}\\% [\/latex] interest, and the investor earns $737.50 annual interest, how much was invested in each account?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134061792\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061793\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134061792-solution\">73<\/a><span class=\"os-divider\">. <\/span>If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134061803\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061804\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>USB-C cables cost $5.96 more than lightning cables at All Bets Are Off Electronics. How much would 6 USB-C cables and 2 lightning cables cost if 5 USB-C cables and 2 lightning cables cost $127.73?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134061810\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061811\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134061810-solution\">75<\/a><span class=\"os-divider\">. <\/span>A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $6,404.40, how many of each type of sneaker were sold?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135363402\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135363403\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span>A concert manager for Red Rocks Ampitheatre counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $5,075 was taken in. How many student tickets and adult tickets were sold?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135363409\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135363410\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135363409-solution\">77<\/a><span class=\"os-divider\">. <\/span>Admission into Elitch Gardens for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and adults, what is the price of the child\u2019s ticket and the price of the adult ticket?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_0bb1dacf-671b-4fe5-a5a4-e67ad9a48d54\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Solve systems of equations by graphing.<\/li>\n<li>Solve systems of equations by substitution.<\/li>\n<li>Solve systems of equations by addition.<\/li>\n<li>Identify inconsistent systems of equations containing two variables.<\/li>\n<li>Express the solution of a system of dependent equations containing two variables.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_1027\" aria-describedby=\"caption-attachment-1027\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1027\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-300x155.webp\" alt=\"\" width=\"300\" height=\"155\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-300x155.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-65x34.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-225x116.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1-350x181.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-1.webp 649w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1027\" class=\"wp-caption-text\">Figure 1. (credit: Thomas S\u00f8renes)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134188850\">A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.<\/p>\n<section id=\"fs-id1165133236395\" data-depth=\"1\">\n<h2 data-type=\"title\">Introduction to Systems of Equations<\/h2>\n<p id=\"fs-id1165137448108\">In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A <strong><span id=\"term-00001\" data-type=\"term\">system of linear equations<\/span><\/strong> consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.<\/p>\n<p id=\"fs-id1165137885053\">In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y &=& 15 \\\\ 3x-y &=& 5 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135596356\">The <em data-effect=\"italics\">solution<\/em> to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2(4)+(7) &=& 15 && \\text{True} \\\\ 3(4)-(7) &=& 5 && \\text{True} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165132912573\">In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A <strong>consistent system<\/strong> of equations has at least one solution. A consistent system is considered to be an <strong>independent system<\/strong> if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a <strong>dependent system<\/strong> if the equations have the same slope and the same <em data-effect=\"italics\">y<\/em>-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.<\/p>\n<p id=\"fs-id1165135262630\">Another type of system of linear equations is an <strong>inconsistent system<\/strong>, which is one in which the equations represent two parallel lines. The lines have the same slope and different <em data-effect=\"italics\">y-<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Types of Linear Systems <\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>There are three types of systems of linear equations in two variables, and three types of solutions.<\/p>\n<ul>\n<li>An <strong>independent system<\/strong> has exactly one solution pair\u00a0[latex](x, y).[\/latex] The point where the two lines intersect is the only solution.<\/li>\n<li>An <strong>inconsistent system<\/strong> has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\n<li>A <strong>dependent system<\/strong> has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\n<\/ul>\n<p>Figure 2 compares the graphical representations of each type of system.<\/p>\n<figure id=\"attachment_1029\" aria-describedby=\"caption-attachment-1029\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1029\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-300x116.webp\" alt=\"\" width=\"300\" height=\"116\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-300x116.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-1024x396.webp 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-768x297.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-65x25.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-225x87.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2-350x135.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-2.webp 1029w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1029\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.<\/strong><\/p>\n<ol>\n<li>Substitute the ordered pair into each equation in the system.<\/li>\n<li>Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Determining Whether an Ordered Pair Is a Solution to a System of Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine whether the ordered pair [latex](5, 1)[\/latex] is a solution to the given system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+3y &=& 8 \\\\ 2x-9 &=& y \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Substitute the ordered pair [latex](5, 1)[\/latex] into both equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} (5)+3(1) &=& 8 \\\\ 8 &=& 8 && \\text{True} \\\\ 2(5)-9 &=& (1) \\\\ 1 &=& 1 && \\text{True} \\end{array}[\/latex]<\/p>\n<p>The ordered pair [latex](5, 1)[\/latex] satisfies both equations, so it is the solution to the system.<\/p>\n<h3>Analysis<\/h3>\n<p>We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines. See Figure 3.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1030\" aria-describedby=\"caption-attachment-1030\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1030\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-300x298.webp\" alt=\"\" width=\"300\" height=\"298\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-300x298.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-65x65.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-225x224.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3-350x348.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-3.webp 376w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1030\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine whether the ordered pair [latex](8, 5)[\/latex] is a solution to the following system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 5x-4y &=& 20 \\\\ 2x+1 &=& 3y \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134573832\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Systems of Equations by Graphing<\/h2>\n<p id=\"fs-id1165137911750\">There are multiple methods of solving systems of linear equations. For a <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">system of linear equations<\/span> in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Solving a System of Equations in Two Variables by Graphing<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations by graphing. Identify the type of system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y &=& -8 \\\\ x-y &=& -1 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Solve the first equation for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y &=& -8 \\\\ y &=& -2x-8\\end{array}[\/latex]<\/p>\n<p>Solve the second equation for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-y &=& -1 \\\\ y &=& x+1 \\end{array}[\/latex]<\/p>\n<p>Graph both equations on the same set of axes as in Figure 4.<\/p>\n<figure id=\"attachment_1031\" aria-describedby=\"caption-attachment-1031\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1031\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-300x240.webp\" alt=\"\" width=\"300\" height=\"240\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-300x240.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-65x52.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-225x180.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4-350x280.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-4.webp 467w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1031\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>The lines appear to intersect at the point [latex](-3, -2).[\/latex] We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2(-3)+(-2) &=& -8 \\\\ -8 &=& -8 && \\text{True} \\\\ (-3)-(-2) &=& -1 \\\\ -1 &=& -1 && \\text{True} \\end{array}[\/latex]<\/p>\n<p>The solution to the system is the ordered pair [latex](-3, -2),[\/latex] so the system is independent.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations by graphing.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x-5y &=& -25 \\\\ -4x+5y &=& 35 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">Q&amp;A<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165137443116\"><strong>Q: Can graphing be used if the system is inconsistent or dependent?<\/strong><\/p>\n<p id=\"fs-id1165133015238\"><em data-effect=\"italics\">A: Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135192374\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Systems of Equations by Substitution<\/h2>\n<p id=\"fs-id1165134361391\">Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the <strong>substitution method<\/strong>, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a system of two equations in two variables, solve using the substitution method.<\/strong><\/p>\n<ol>\n<li>Solve one of the two equations for one of the variables in terms of the other.<\/li>\n<li>Substitute the expression for this variable into the second equation, then solve for the remaining variable.<\/li>\n<li>Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.<\/li>\n<li>Check the solution in both equations.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Solving a System of Equations in Two Variables by Substitution<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations by substitution.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -x+y &=& -5 \\\\ 2x-5y &=& 1 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, we will solve the first equation for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -x+y &=& -5 \\\\ y &=& x-5 \\end{array}[\/latex]<\/p>\n<p>Now we can substitute the expression [latex]x-5[\/latex] for y in the second equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x-5y &=& 1 \\\\ 2x-5(x-5) &=& 1 \\\\ 2x-5x+25 &=& 1 \\\\ -3x &=& -24 \\\\ x &=& 8 \\end{array}[\/latex]<\/p>\n<p>Now, we substitute [latex]x=8[\/latex] into the first equation and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -(8)+y &=& -5 \\\\ y &=& 3 \\end{array}[\/latex]<\/p>\n<p>Our solution is [latex](8, 3).[\/latex]<\/p>\n<p>Check the solution by substituting [latex](8, 3)[\/latex] into both equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -x+y &=& -5 \\\\ -(8)+(3) &=& -5 && \\text{True} \\\\ 2x-5y &=& 1 \\\\ 2(8)-5(3) &=& 1 && \\text{True} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations by substitution.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x &=& y+3 \\\\ 4 &=& 3x-2y \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"eip-id3042099\"><strong>Q: Can the substitution method be used to solve any linear system in two variables?<\/strong><\/p>\n<p id=\"fs-id1165132973467\"><em data-effect=\"italics\">A: Yes, but the method works best if one of the equations contains a coefficient of 1 or \u20131 so that we do not have to deal with fractions.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165132972949\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Systems of Equations in Two Variables by the Addition Method<\/h2>\n<p id=\"fs-id1165131968519\">A third method of <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">solving systems of linear equations<\/span> is the <strong><span id=\"term-00010\" data-type=\"term\">addition method<\/span><\/strong>. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a system of equations, solve using the addition method.<\/strong><\/p>\n<ol>\n<li>Write both equations with <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-variables on the left side of the equal sign and constants on the right.<\/li>\n<li>Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.<\/li>\n<li>Solve the resulting equation for the remaining variable.<\/li>\n<li>Substitute that value into one of the original equations and solve for the second variable.<\/li>\n<li>Check the solution by substituting the values into the other equation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Solving a System by the Addition Method<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the given system of equations by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+2y &=& -1 \\\\ -x+y &=& 3 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Both equations are already set equal to a constant. Notice that the coefficient of <em>x<\/em> in the second equation, \u20131, is the opposite of the coefficient of <em>x<\/em> in the first equation, 1. We can add the two equations to eliminate <em>x<\/em> without needing to multiply by a constant.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+2y &=& -1 \\\\ -x+y &=& 3 \\\\ \\hline 3y &=& 2 \\end{array}[\/latex]<\/p>\n<p>Now that we have eliminated x, we can solve the resulting equation for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3y &=& 2 \\\\ y &=& \\frac{2}{3} \\end{array}[\/latex]<\/p>\n<p>Then, we substitute this value for <em>y<\/em> into one of the original equations and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -x+y &=& 3 \\\\ -x+\\frac{2}{3} &=& 3 \\\\ -x &=& 3-\\frac{2}{3} \\\\ -x &=& \\frac{7}{3} \\\\ x &=& -\\frac{7}{3} \\end{array}[\/latex]<\/p>\n<p>The solution to this system is [latex]\\left(-\\frac{7}{3}, \\frac{2}{3}\\right).[\/latex]<\/p>\n<p>Check the solution in the first equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+2y &=& -1 \\\\ \\left(-\\frac{7}{3}\\right)+2\\left(\\frac{2}{3}\\right) &=& -1 \\\\ -\\frac{7}{3}+\\frac{4}{3} &=& -1 \\\\ -\\frac{3}{3} &=& -1 \\\\ -1 &=& -1 && \\text{True} \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We gain an important perspective on systems of equations by looking at the graphical representation. See Figure 5 to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution.<\/p>\n<figure id=\"attachment_1032\" aria-describedby=\"caption-attachment-1032\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1032\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-300x236.webp\" alt=\"\" width=\"300\" height=\"236\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-300x236.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-65x51.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-225x177.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5-350x275.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-5.webp 476w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1032\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165137760592\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135190076\" data-type=\"commentary\">\n<div id=\"Figure_09_01_004\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Using the Addition Method When Multiplication of One Equation Is Required<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the given system of equations by the addition method.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3x+5y &=& -11 \\\\ x-2y &=& 11 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Adding these equations as presented will not eliminate a variable. However, we see that the first equation has 3x in it and the second equation has <em>x<\/em>. So if we multiply the second equation by -3, the <em data-effect=\"italics\">x<\/em>-terms will add to zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2y &=& 11 \\\\ -3(x-2y) &=& -3(11) && \\text{Multiply both sides by } -3. \\\\ -3x+6y &=& -33 && \\text{Use the distributive property.} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165134567942\">Now, let\u2019s add them.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3x+5y &=& -11 \\\\ -3x+6y &=& -33 \\\\ \\hline 11y &=& -44 \\\\ y &=& -4 \\end{array}[\/latex]<\/p>\n<p>For the last step, we substitute [latex]y=-4[\/latex] into one of the original equations and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 3x+5y &=& -11 \\\\ 3x+5(-4) &=& -11 \\\\ 3x-20 &=& -11 \\\\ 3x &=& 9 \\\\ x &=& 3 \\end{array}[\/latex]<\/p>\n<p>Our solution is the ordered pair [latex](3, -4).[\/latex] See Figure 6. Check the solution in the original second equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x-2y &=& 11 \\\\ (3)-2(-4) &=& 3+8 \\\\ 11 &=& 11 && \\text{True} \\end{array}[\/latex]<\/p>\n<figure id=\"attachment_1033\" aria-describedby=\"caption-attachment-1033\" style=\"width: 285px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1033\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-285x300.webp\" alt=\"\" width=\"285\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-285x300.webp 285w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-65x68.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-225x237.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6-350x369.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-6.webp 357w\" sizes=\"auto, (max-width: 285px) 100vw, 285px\" \/><figcaption id=\"caption-attachment-1033\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system of equations by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x-7y &=& 2 \\\\ 3x+y &=& -20 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Using the Addition Method When Multiplication of Both Equations Is Required<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the given system of equations in two variables by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}2x+3y &=& -16 \\\\ 5x-10y &=& 30 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>One equation has 2x and the other has 5x. The least common multiple is 10x so we will have to multiply both equations by a constant in order to eliminate one variable. Let\u2019s eliminate <em>x<\/em> by multiplying the first equation by [latex]-5[\/latex] and the second equation by [latex]2.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -5(2x+3y) &=& -5(-16) \\\\ -10x-15y &=& 80 \\\\ 2(5x-10y) &=& 2(30) \\\\ 10x-20y &=& 60 \\end{array}[\/latex]<\/p>\n<p>Then, we add the two equations together.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -10x-15y &=& 80 \\\\ 10x-20y &=& 60 \\\\ \\hline -35y &=& 140 \\\\ y &=& -4 \\end{array}[\/latex]<\/p>\n<p>Substitute [latex]y=-4[\/latex] into the original first equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+3(-4) &=& -16 \\\\ 2x-12 &=& -16 \\\\ 2x &=& -4 \\\\ x &=& -2 \\end{array}[\/latex]<\/p>\n<p>The solution is [latex]- (-2, -4).[\/latex] Check it in the other equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 5x-10y &=& 30 \\\\ 5(-2)-10(-4) &=& 30 \\\\ -10+40 &=& 30 \\\\ 30 &=& 30 \\end{array}[\/latex]<\/p>\n<p>See Figure 7.<\/p>\n<figure id=\"attachment_1034\" aria-describedby=\"caption-attachment-1034\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1034\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-300x230.webp\" alt=\"\" width=\"300\" height=\"230\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-300x230.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-65x50.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-225x173.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7-350x269.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-7.webp 509w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1034\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Using the Addition Method in Systems of Equations Containing Fractions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the given system of equations in two variables by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} \\frac{x}{3}+\\frac{y}{6} &=& 3 \\\\ \\frac{x}{2}-\\frac{y}{4} &=& 1 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First clear each equation of fractions by multiplying both sides of the equation by the least common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 6\\left(\\frac{x}{3}+\\frac{y}{6}\\right) &=& 6(3) \\\\ 2x+y &=& 18 \\\\ 4\\left(\\frac{x}{2}-\\frac{y}{4}\\right) &=& 4(1) \\\\ 2x-y &=& 4 \\end{array}[\/latex]<\/p>\n<p>Now multiply the second equation by [latex]-1[\/latex] so that we can eliminate the <em data-effect=\"italics\">x<\/em>-variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -1(2x-y) &=& -1(4) \\\\ -2x+y &=& -4 \\end{array}[\/latex]<\/p>\n<p>Add the two equations to eliminate the <em data-effect=\"italics\">x<\/em>-variable and solve the resulting equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+y &=& 18 \\\\ -2x+y &=& -4 \\\\ \\hline 2y &=& 14 \\\\ y &=& 7 \\end{array}[\/latex]<\/p>\n<p>Substitute [latex]y=7[\/latex] into the first equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+(7) &=& 18 \\\\ 2x &=& 11 \\\\ x &=& \\frac{11}{2} \\\\ &=& 5.5 \\end{array}[\/latex]<\/p>\n<p>The solution is [latex]\\left(\\frac{11}{2}, 7\\right).[\/latex] Check it in the other equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} \\frac{x}{2}-\\frac{y}{4} &=& 1 \\\\ \\frac{\\frac{11}{2}}{2}-\\frac{7}{4} &=& 1 \\\\ \\frac{11}{4}-\\frac{7}{4} &=& 1 \\\\ \\frac{4}{4} &=& 1 \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system of equations by addition.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2x+3y &=& 8 \\\\ 3x+5y &=& 10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135462954\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying Inconsistent Systems of Equations Containing Two Variables<\/h2>\n<p id=\"fs-id1165135462960\">Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">inconsistent system<\/span> consists of parallel lines that have the same slope but different <em>y<\/em>-intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex]12=0.[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Solving an Inconsistent System of Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x &=& 9-2y \\\\ x+2y &=& 13 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can approach this problem in two ways. Because one equation is already solved for <em>x<\/em>, the most obvious step is to use substitution.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+2y &=& 13 \\\\ (9-2y)+2y &=& 13 \\\\ 9+0y &=& 13 \\\\ 9 &=& 13 \\end{array}[\/latex]<\/p>\n<p>Clearly, this statement is a contradiction because [latex]9\\not=13.[\/latex] Therefore, the system has no solution.<\/p>\n<p>The second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x &=& 9-2y \\\\ 2y &=& -x+9 \\\\ y &=& -\\frac{1}{2}x+\\frac{9}{2} \\end{array}[\/latex]<\/p>\n<p>We then convert the second equation expressed to slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+2y &=& 13 \\\\ 2y &=& -x+13 \\\\ y &=& -\\frac{1}{2}x+\\frac{13}{2} \\end{array}[\/latex]<\/p>\n<p>Comparing the equations, we see that they have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. Therefore, the lines are parallel and do not intersect.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& -\\frac{1}{2}x+\\frac{9}{2} \\\\ y &=& -\\frac{1}{2}+\\frac{13}{2} \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure 8.<\/p>\n<figure id=\"attachment_1037\" aria-describedby=\"caption-attachment-1037\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1037\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-300x213.webp\" alt=\"\" width=\"300\" height=\"213\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-300x213.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-65x46.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-225x159.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8-350x248.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-8.webp 525w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1037\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165137701571\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165134116945\" data-type=\"commentary\">\n<div id=\"Figure_09_01_007\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 2y-2x &=& 2 \\\\ 2y-2x &=& 6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135189716\" data-depth=\"1\">\n<h2 data-type=\"title\">Expressing the Solution of a System of Dependent Equations Containing Two Variables<\/h2>\n<p id=\"fs-id1165133200582\">Recall that a <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">dependent system<\/span> of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as [latex]0=0.[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Finding a Solution to a Dependent System of Linear Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a solution to the system of equations using the addition method.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+3y &=& 2 \\\\ 3x+9y &=& 6 \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let\u2019s focus on eliminating <em>x<\/em>. If we multiply both sides of the first equation by [latex]-3,[\/latex] then we will be able to eliminate the <em>x<\/em>-variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+3y &=& 2 \\\\ (-3)(x+3y) &=& (-3)(2) \\\\ -3x-9y &=& -6 \\end{array}[\/latex]<\/p>\n<p>Now add the equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} -3x-9y &=& -6 \\\\ + \\ \\ \\ 3x+9y &=& 6 \\\\ \\hline 0 &=& 0 \\end{array}[\/latex]<\/p>\n<p>We can see that there will be an infinite number of solutions that satisfy both equations.<\/p>\n<h3>Analysis<\/h3>\n<p>If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let\u2019s look at what happens when we convert the system to slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} x+3y &=& 2 \\\\ 3y &=& -x+2 \\\\ y &=& -\\frac{1}{3}x+\\frac{2}{3} \\\\ 3x+9y &=& 6 \\\\ 9y &=& -3x+6 \\\\ y &=& -\\frac{3}{9}x+\\frac{6}{9} \\\\ y &+& -\\frac{1}{3}x+\\frac{2}{3} \\end{array}[\/latex]<\/p>\n<p>See Figure 9. Notice the results are the same. The general solution to the system is [latex]\\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right).[\/latex]<\/p>\n<figure id=\"attachment_1039\" aria-describedby=\"caption-attachment-1039\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1039\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-300x254.webp\" alt=\"\" width=\"300\" height=\"254\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-300x254.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-65x55.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-225x191.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9-350x297.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-9.webp 441w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1039\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165133299011\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165133188520\" data-type=\"commentary\">\n<div id=\"Figure_09_01_008\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the following system of equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y-2x &=& 5 \\\\ -3y+6x &=& -15 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135559421\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Systems of Equations to Investigate Profits<\/h2>\n<p id=\"fs-id1165135559426\">Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer\u2019s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation [latex]R=xp,[\/latex] where [latex]x=[\/latex] quantity and [latex]p=[\/latex] price. The revenue function is shown in orange in Figure 10.<\/p>\n<p id=\"fs-id1165135459707\">The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in Figure 10. The x-axis represents quantity in hundreds of units. The <em data-effect=\"italics\">y<\/em>-axis represents either cost or revenue in hundreds of dollars.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"Figure_09_01_009\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_1040\" aria-describedby=\"caption-attachment-1040\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1040\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-300x212.webp\" alt=\"\" width=\"300\" height=\"212\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-300x212.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-65x46.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-225x159.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10-350x247.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-10.webp 612w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1040\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135449745\">The point at which the two lines intersect is called the <strong>break-even point<\/strong>. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.<\/p>\n<p id=\"fs-id1165135449753\">The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The <strong>profit function<\/strong> is the revenue function minus the cost function, written as [latex]P(x)=R(x)-C(x).[\/latex] Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Finding the Break-Even Point and the Profit Function Using Substitution<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the cost function [latex]C(x)=0.85x+35,000[\/latex] and the revenue function [latex]R(x)=1.55x,[\/latex] find the break-even point and the profit function.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Write the system of equations using y to replace function notation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} y &=& 0.85x+35,000 \\\\ y &=& 1.55x \\end{array}[\/latex]<\/p>\n<p>Substitute the expression [latex]0.85x+35,000[\/latex] from the first equation into the second equation and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 0.85x+35,000 &=& 1.55x \\\\ 35,000 &=& 0.7x \\\\ 50,000 &=& x \\end{array}[\/latex]<\/p>\n<p>Then, we substitute [latex]x=50,000[\/latex] into either the cost function or the revenue function.<\/p>\n<p style=\"text-align: center;\">[latex]1.55(50,000)=77,500[\/latex]<\/p>\n<p>The break-even point is [latex](50,000, 77,500).[\/latex]<\/p>\n<p>The profit function is found using the formula [latex]P(x)=R(x)-C(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} P(x) &=& 1.55x-(0.85x+35,000) \\\\ &=& 0.7x-35,000 \\end{array}[\/latex]<\/p>\n<p>The profit function is [latex]P(x)=0.7x-35,000.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11.<\/p>\n<figure id=\"attachment_1041\" aria-describedby=\"caption-attachment-1041\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1041\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-300x211.webp\" alt=\"\" width=\"300\" height=\"211\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-300x211.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-65x46.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-225x158.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11-350x246.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-11.webp 562w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1041\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<p>We see from the graph in Figure 12 that the profit function has a negative value until [latex]x=50,000,[\/latex] when the graph crosses the <em data-effect=\"italics\">x<\/em>-axis. Then, the graph emerges into positive <em data-effect=\"italics\">y<\/em>-values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.<\/p>\n<figure id=\"attachment_1042\" aria-describedby=\"caption-attachment-1042\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1042\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-300x168.webp\" alt=\"\" width=\"300\" height=\"168\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-300x168.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-65x36.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-225x126.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12-350x196.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.1-fig-12.webp 725w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1042\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165134040438\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165131987013\" data-type=\"commentary\">\n<div id=\"Figure_09_01_010\" class=\"os-figure\"><\/div>\n<div id=\"Figure_09_01_011\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Writing and Solving a System of Equations in Two Variables<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The cost of a ticket to Elitch Gardens is for [latex]\\$25.00[\/latex] children and [latex]\\$50.00[\/latex] for adults. On a certain day, attendance at Elitch Gardens is [latex]2,000[\/latex] and the total gate revenue is [latex]\\$70,000.[\/latex]How many children and how many adults bought tickets?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Let <em data-effect=\"italics\">c<\/em> = the number of children and <em data-effect=\"italics\">a<\/em> = the number of adults in attendance.<\/p>\n<p>The total number of people is 2,000. We can use this to write an equation for the number of people at the circus that day.<\/p>\n<p style=\"text-align: center;\">[latex]c+a=2,000[\/latex]<\/p>\n<p>The revenue from all children can be found by multiplying [latex]\\$25.00[\/latex] by the number of children, [latex]25c.[\/latex] The revenue from all adults can be found by multiplying [latex]\\$50.000[\/latex] by the number of adults, [latex]50a.[\/latex] The total revenue is [latex]\\$70,000.[\/latex] We can use this to write an equation for the revenue.<\/p>\n<p style=\"text-align: center;\">[latex]25c+50a=70,000[\/latex]<\/p>\n<p>We now have a system of linear equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} c+a &=& 2,000 \\\\ 25c+50a &=& 70,000 \\end{array}[\/latex]<\/p>\n<p>In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either <em>c<\/em> or <em>a<\/em>. We will solve for <em>a<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} c+a &=& 2,000 \\\\ a &=& 2,000-c \\end{array}[\/latex]<\/p>\n<p>Substitute the expression [latex]2,000-c[\/latex] in the second equation for <em>a<\/em> and solve for <em>c<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 25x+50(2,000-c) &=& 70,000 \\\\ 25c+100,000-50c &=& 70,000 \\\\ -25c &=& -30,000 \\\\ c &=& 1,200 \\end{array}[\/latex]<\/p>\n<p>Substitute [latex]c=1,200[\/latex] into the first equation to solve for <em>a<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} 1,200+a &=& 2,000 \\\\ a &=& 800 \\end{array}[\/latex]<\/p>\n<p>We find that 1,200 children and 800 adults bought tickets to Elitch Gardens that day.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Meal tickets at Elitch Gardens cost [latex]\\$4.00[\/latex] for children and [latex]\\$12.00[\/latex] for adults. If 1,650 meal tickets were bought for a total of [latex]\\$14,200[\/latex] how many children and how many adults bought meal tickets?<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with systems of linear equations.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=HxhacvH49o8\">Solving Systems of Equations Using Substitution<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=ova8GSmPV4o\">Solving Systems of Equations Using Elimination<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=GTolISAXveg\">Applications of Systems of Equations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.1 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135186955\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135186958\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165133393400\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133393402\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133393400-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can a system of linear equations have exactly two solutions? Explain why or why not.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133393410\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133393411\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company\u2019s profit margins.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133393418\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133393419\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133393418-solution\">3<\/a><span class=\"os-divider\">. <\/span>If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133305366\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133305367\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133305372\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133305374\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133305372-solution\">5<\/a><span class=\"os-divider\">. <\/span>Given a system of equations, explain at least two different methods of solving that system.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133232429\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165133232434\">For the following exercises, determine whether the given ordered pair is a solution to the system of equations.<\/p>\n<div id=\"fs-id1165133232439\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133232440\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll}5x-y &=& 4 \\\\ x+6y &=& 2 \\end{array}[\/latex] and [latex](4, 0)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132934456\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132934457\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132934456-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -3x-5y &=& 13 \\\\ -x+4y &=& 10 \\end{array}[\/latex] and [latex](-6, 1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133294269\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133294270\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x+7y &=& 1 \\\\ 2x+4y &=& 0 \\end{array}[\/latex] and [latex](2, 3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135255522\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135255523\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135255522-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -2x+5y &=& 7 \\\\ 2x+9y &=& 7 \\end{array}[\/latex] and [latex](-1, 1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134350008\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134350009\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+8y &=& 43 \\\\ 3x-2y &=& -1 \\end{array}[\/latex] and [latex](3, 5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134556458\">For the following exercises, solve each system by substitution.<\/p>\n<div id=\"fs-id1165134556461\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134556462\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134556461-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+3y &=& 5 \\\\ 2x+3y &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137666024\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137666025\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-2y &=& 18 \\\\ 5x+10y &=& -10 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134254239\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134254240\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134254239-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 4x+2y &=& -10\\\\ 3x+9y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133265018\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133265020\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2x+4y &=& -3.8 \\\\ 9x-5y &=& 1.3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134171257\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134171258\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134171257-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -2x+3y &=& 1.2 \\\\ -3x-6y &=& 1.8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137696015\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137696016\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x-0.2y &=& 1 \\\\ -10x+2y &=& 5 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133447806\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133447807\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133447806-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x+5y &=& 9 \\\\ 30x+50y &=& -90 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133370753\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133370754\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -3x+y &=& 2 \\\\ 12x-4y &=& -8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134162139\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134162140\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134162139-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{2}x+\\frac{1}{3}y &=& 16 \\\\ \\frac{1}{6}x+\\frac{1}{4}y &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134319658\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134319659\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -\\frac{1}{4}x+\\frac{3}{2}y &=& 11 \\\\ -\\frac{1}{8}x+\\frac{1}{3}y &=& 3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134568907\">For the following exercises, solve each system by addition.<\/p>\n<div id=\"fs-id1165134568910\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134568911\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134568910-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -2x+5y &=& -42 \\\\ 7x+2y &=& 30 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135407254\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135407255\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 6x-5y &=& -34 \\\\ 2x+6y &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134554922\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134554923\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134554922-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-y &=& -2.6 \\\\ -4x-6y &=& 1.4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134357764\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134357765\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 7x-2y &=& 3 \\\\ 4x+5y &=& 3.25 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132919470\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132919471\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132919470-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x+2y &=& -1 \\\\ 5x-10y &=& 6 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133109448\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133109449\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 7x+6y &=& 2 \\\\ -28x-24y &=& -8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132924229\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132924230\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132924229-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{5}{6}x+\\frac{1}{4}y &=& 0 \\\\ \\frac{1}{8}x-\\frac{1}{2}y &=& -\\frac{43}{120} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132931032\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132931033\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{3}x+\\frac{1}{9}y &=& \\frac{2}{9} \\\\ -\\frac{1}{2}x+\\frac{4}{5}y &=& -\\frac{1}{3} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135183918\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135183919\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135183918-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -0.2x+0.4y &=& 0.6 \\\\ x-2y &=& -3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131991291\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131991292\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -0.1x+0.2y &=& 0.6 \\\\ 5x-10y &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134278608\">For the following exercises, solve each system by any method.<\/p>\n<div id=\"fs-id1165134278611\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134278612\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134278611-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x+9y &=& 16 \\\\ x+2y &=& 4 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133172708\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133172709\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 6x-8y &=& -0.6 \\\\ 3x+2y &=& 0.9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134430261\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134430262\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134430261-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 5x-2y &=& 2.25 \\\\ 7x-4y &=& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135705758\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135705759\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x-\\frac{5}{12}y &=& -\\frac{55}{12} \\\\ -6x+\\frac{5}{2}y &=& \\frac{55}{2} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134329672\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134329673\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134329672-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 7x-4y &=& \\frac{7}{6} \\\\ 2x+4y &=& \\frac{1}{3} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134272660\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134272661\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x+6y &=& 11 \\\\ 2x+4y &=& 9 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132939103\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132939104\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132939103-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{7}{3}x-\\frac{1}{6}y &=& 2 \\\\ -\\frac{21}{6}x+\\frac{3}{12}y &=& -3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131893549\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131893550\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} \\frac{1}{2}x+\\frac{1}{3}y &=& \\frac{1}{3} \\\\ \\frac{3}{2}x+\\frac{1}{4}y &=& -\\frac{1}{8} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135299787\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135299788\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135299787-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 2.2x+1.3y &=& -0.1 \\\\ 4.2x+4.2y &=& 2.1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135702539\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135702540\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.1x+0.2y &=& 2 \\\\ 0.35x-0.3y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135518201\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135518207\">For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.<\/p>\n<div id=\"fs-id1165135518212\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135518213\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135518212-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-y &=& 0.6 \\\\ x-2y &=& 1.3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165134279206\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134279207\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -x+2y &=& 4 \\\\ 2x-4y &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132957245\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132957246\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132957245-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+2y &=& 7 \\\\ 2x+6y &=& 12 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133074995\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133074996\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-5y &=& 7 \\\\ x-2y &=& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133243607\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133243608\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133243607-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 3x-2y &=& 5 \\\\ -9x+6y &=& -15 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165133290332\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135388295\">For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.<\/p>\n<div id=\"fs-id1165135388300\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135388301\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.1x+0.2y &=& 0.3 \\\\ -0.3x+0.5y &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137836999\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137837000\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137836999-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -0.01x+0.12y &=& 0.62 \\\\ 0.15x+0.20y &=& 0.52 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134035993\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134035994\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.5x+0.3y &=& 4 \\\\ 0.25x-0.9y &=& 0.46 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134361357\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134361358\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134361357-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} 0.15x+0.27y &=& 0.39 \\\\ -0.34x+0.56y &=& 1.8 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134485557\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134485558\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} -0.71x+0.92y &=& 0.13 \\\\ 0.83x+0.05y &=& 2.1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134113096\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165134113102\">For the following exercises, solve each system in terms of [latex]A, B, C, D, E,[\/latex] and [latex]F[\/latex] where [latex]A-F[\/latex] are nonzero numbers. Note that [latex]A\\not=B[\/latex] and [latex]AE\\not=BD.[\/latex]<\/p>\n<div id=\"fs-id1165132923031\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132923032\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132923031-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+y &=& A \\\\ x-y &=& B \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135621951\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134385523\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} x+Ay &=& 1 \\\\ x+By &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132972781\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132972782\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165132972781-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} Ax+y &=& 0 \\\\ Bx+y &=& 0 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137848824\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137848825\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} Ax+By &=& C \\\\ x+y &=& 1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133210727\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133210728\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133210727-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]\\begin{array}{rll} Ax+By &=& C \\\\ Dx+Ey &=& F \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135702580\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1165135702585\">For the following exercises, solve for the desired quantity.<\/p>\n<div id=\"fs-id1165135702588\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135702589\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>A stuffed animal business has a total cost of production [latex]C=12x+30[\/latex] and a revenue function [latex]R=20x.[\/latex] Find the break-even point.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135486976\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135486977\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135486976-solution\">57<\/a><span class=\"os-divider\">. <\/span>An Ethiopian restaurant has a cost of production [latex]C(x)=11x+120[\/latex] and a revenue function [latex]R(x)=5x.[\/latex] When does the company start to turn a profit?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133315062\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133315063\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>A cell phone factory has a cost of production [latex]C(x)=150x+10,000[\/latex] and a revenue function [latex]R(x)=200x.[\/latex] What is the break-even point?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133249125\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133249126\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133249125-solution\">59<\/a><span class=\"os-divider\">. <\/span>A musician charges [latex]C(x)=64x+20,000[\/latex] where <em>x<\/em> is the total number of attendees at the concert. Red Rocks Ampitheatre charges $80 per ticket. After how many people buy tickets does Red Rocks Ampitheatre break even, and what is the value of the total tickets sold at that point?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134231536\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134231537\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>A guitar factory has a cost of production [latex]C(x)=75x+50,000.[\/latex] If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165131863006\">For the following exercises, use a system of linear equations with two variables and two equations to solve.<\/p>\n<div id=\"fs-id1165131863011\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131863012\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131863011-solution\">61<\/a><span class=\"os-divider\">. <\/span>Find two numbers whose sum is 28 and difference is 13.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131863020\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131863021\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131863026\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131863027\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131863026-solution\">63<\/a><span class=\"os-divider\">. <\/span>The startup cost for a restaurant in Aurora, CO, is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199413\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199414\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199420\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199422\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134199420-solution\">65<\/a><span class=\"os-divider\">. <\/span>A total of 1,595 first- and second-year CCA students gathered at a pep rally. The number of first-years exceeded the number of second-years by 15. How many students from each year group were in attendance?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199432\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199433\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>276 CCA students enrolled in a College Algebra class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199439\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199440\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134199439-solution\">67<\/a><span class=\"os-divider\">. <\/span>There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134254378\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134254379\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>A jeep and a pickup truck enter I 70 running east-west at the same exit heading in opposite directions. The jeep entered I 70 30 minutes before the pickup did, and traveled 7 mph slower than the pickup. After 2 hours from the time the pickup entered I 70, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control and retained the same speed.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134254387\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134254388\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134254387-solution\">69<\/a><span class=\"os-divider\">. <\/span>If a CCA Chemistry Professor mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many<br \/>\ngallons of 10% and 60% solutions were mixed?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134254398\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134254399\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>An investor earned triple the profits of what they earned last year. If they made $500,000.48 total for both years, how much did the investor earn in profits each year?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133227637\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133227638\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133227637-solution\">71<\/a><span class=\"os-divider\">. <\/span>An investor invested 1.1 million dollars into two land investments. On the first investment, the Aurora Highlands, her return was a 110% increase on the money she invested. On the second investment, Westerley Creek, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133227650\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133227651\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the other that pays [latex]2 \\frac{7}{8}\\%[\/latex] interest, and the investor earns $737.50 annual interest, how much was invested in each account?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134061792\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061793\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134061792-solution\">73<\/a><span class=\"os-divider\">. <\/span>If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134061803\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061804\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>USB-C cables cost $5.96 more than lightning cables at All Bets Are Off Electronics. How much would 6 USB-C cables and 2 lightning cables cost if 5 USB-C cables and 2 lightning cables cost $127.73?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134061810\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061811\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134061810-solution\">75<\/a><span class=\"os-divider\">. <\/span>A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $6,404.40, how many of each type of sneaker were sold?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135363402\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135363403\" data-type=\"problem\">\n<p><span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span>A concert manager for Red Rocks Ampitheatre counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $5,075 was taken in. How many student tickets and adult tickets were sold?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135363409\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135363410\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135363409-solution\">77<\/a><span class=\"os-divider\">. <\/span>Admission into Elitch Gardens for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and adults, what is the price of the child\u2019s ticket and the price of the adult ticket?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-212","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/212","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":15,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/212\/revisions"}],"predecessor-version":[{"id":1681,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/212\/revisions\/1681"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/212\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=212"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=212"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=212"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}